University of Toronto
Department of Economics
September 25, 2007 By Victor Aguirregabiria
Comment: Identification of a Simple Dynamic Discrete Game
under Rationalizability
Comment: Identi
fi
cation of a Simple Dynamic Discrete
Game under Rationalizability
∗
Victor Aguirregabiria
†University of Toronto
September 22, 2007
Abstract
This paper studies the identification power ofrationalizability in a simple dynamic discrete game model. The paper extends to dynamic games some of the results in Aradillas-Lopez and Tamer (2007). The most commonly used equilibrium concept in empirical applications of dynamic games is Markov Perfect Equilibrium (MPE). I study the identification of structural parameters when we replace the MPE assumption with weaker conditions such as rational behavior or rationalizability. I present identification results for a simple dynamic game of market entry-exit with two players. Under the assumption of level-2 rationalizability (i.e., players are rational and they know that they are rational), exclusion restrictions and large-support conditions on the exoge-nous explanatory variables are sufficient for point-identification of all the structural parameters. Though the model is fully parametric, the key identifying assumptions are nonparametric in nature and it seems that these identification results might be extended to a semiparametric version of the model.
Keywords: Identification, Empirical dynamic discrete games, rational behavior, ra-tionalizability.
∗Comment on the paper"The Identification Power of Equilibrium in Games," by Andres Aradillas-Lopez
and Elie Tamer, invited paper for the annual conference of theJournal of Business and Economics Statistics.
Salt Lake City. July 2007.
†Address: Department of Economics. Unversity of Toronto. 150 St. George Street, Toronto, Ontario
1
Introduction
Structural econometric models of individual or firm behavior typically assume that agents are rational in the sense that they maximize expected payoffs given their subjective beliefs about uncertain events. Empirical applications of game theoretic models have used stronger assumptions than rationality. Most studies that have estimated games have used the Nash equilibrium concept, or some of its refinements, to explain agents’ strategic behavior. The Nash equilibrium (NE) concept is based on assumptions on players’ knowledge and beliefs which are more restrictive than rationality. Though there is not a set of necessary condi-tions to generate the NE outcome, the set of sufficient conditions includes the assumption that players’ actions are common knowledge.1 This assumption may be too restrictive or unrealistic in some applications. Therefore, it seems relevant to study whether we can iden-tify the parameters of empirical games under weaker conditions than NE. For instance, we would like to know if rationality (together with mutual knowledge of payoffs) is sufficient for identification. It is also relevant to study the identification power of other assumptions on players’ knowledge which are stronger than rationality but weaker than NE, such as com-mon knowledge rationality: i.e., everybody knows that players are rational; everybody knows that everybody knows that players are rational, etc. The solution concepts iterated strict dominance andrationalizability are closely related to the assumption of common knowledge rationality (see chapter 2 in Fudenberg and Tirole, 1991).
The paper by Andres Aradillas-Lopez and Elie Tamer (2007) is thefirst study that deals with these interesting identification issues. The authors study the identification power of rational behavior and rationalizability in three classes of static games which have received significant attention in empirical applications: binary choice games, with complete and with incomplete information, and auction games under the independent private values paradigm. Their paper contributes to the literature on identification of incomplete econometric
mod-1Aumann and Brandenburger (1995) have derived sufficient conditions on players’ knowledge and beliefs
to generate the NE as an outcome of a game. They show that mutual knowledge of the payofffunctions and of rationality, and common knowledge of the conjectures (actions), imply that the conjectures form a NE.
els, i.e., models which do not provide unique predictions on the distribution of endogenous variables (see also Tamer, 2003, and Haile and Tamer, 2003). Aradillas-Lopez and Tamer’s paper shows that standard exclusion restrictions and large-support conditions are sufficient to identify structural parameters despite the non-uniqueness of the model predictions. Note that, though structural parameters can be point-identified, when we use the estimated model to perform counterfactual experiments we have that players’ behavior under the counterfac-tual scenario is not point-identified. This problem also appears in models with multiple equilibria. However, a nice feature of Aradillas-Lopez and Tamer’srationalizability approach
is that, at least for the class of models that they consider, it is very simple to obtain bounds for the model the predictions on players’ behavior.
The main purpose of this paper is to study the identification power of rational behavior and rationalizability in a class of empirical games that has not been analyzed in Aradillas-Lopez and Tamer’s paper: dynamic discrete games. Dynamic discrete games are of interest in economic applications where agents interact over several periods of time and make decisions that affect both current and future payoffs. The most commonly used equilibrium concept in applications of dynamic games is Markov Perfect Equilibrium (MPE). As in the case of NE, the concept of MPE is based on the assumption that players’ strategy functions are common knowledge. Here I study the identification of structural parameters when we replace the MPE assumption with weaker conditions such as rational behavior or rationalizability.
I present identification results for a simple dynamic game of market entry-exit with two players. For this simple model the results are similar to the ones in Aradillas-Lopez and Tamer’s paper for static games of incomplete information. Under the assumption of level-2 rationalizability (i.e., players are rational and they know that they are rational), exclusion restrictions and large-support conditions on the exogenous explanatory variables are sufficient for point-identification of all the structural parameters. Though the model is fully parametric, the key identifying assumptions are nonparametric in nature and it seems that these identification results might be extended to a semiparametric version of the model.
2
Dynamic discrete games
2.1
Model and assumptions
There are two firms which decide whether to operate or not in a market. I use the indexes
i∈{1,2} andj ∈{1,2}to represent a firm and its opponent, respectively. Time is discrete an indexed byt. LetYit ∈{0,1} be the indicator of the event "firmiis active in the market
at period t". Every period t the two firms decide simultaneously whether to be active in the market or not. A firm makes this decision to maximize its expected and discounted profits Et(
P∞
s=0δ
s
Πi,t+s), where δ ∈(0,1)is the discount factor and Πit is the firm’s profit
at period t. The decision to be active in the market has implications not only on a firm’s current profits but also on its expected future profits. More specifically, there is an entry cost that should be paid only if a currently active firm was not active at previous period (Yit = 1andYit−1 = 0). Therefore, the lagged entry decisionYit−1 affects current profits and
the model is dynamic. The one-period profit function is:
Πit= ⎧ ⎨ ⎩ Ziηi+γiYit−1 +αiYjt−εit if Yit = 1 0 if Yit = 0 (1)
Yjt represents the opponent’s decision. ηi, γi and αi are parameters. The parameter γi
represents firm i’s entry cost. The parameter αi ≤ 0 captures the competitive effect. The
variable εit is private information of firm i. For the sake of simplicity, I assume that the
exogenous market andfirm characteristics in Z1 andZ2 are constant over time. The vector
of parameters θ≡{ηi,γi, αi :i= 1,2} and the vector of variables Z ≡(Z1, Z2)are common
knowledge. The variables ε1t and ε2t are independent ofZ, independent of each other, and
independently and identically distributed over time. Their distribution functions, H1 and
H2, are absolutely continuous and strictly increasing with respect to the Lebesgue measure
2.2
Rational behavior and rationalizability
The literature on estimation of dynamic discrete games has used the concept of Markov Perfect Equilibrium (MPE). This type of equilibrium assumes that: (1) players’ strategy functions depend only on payoff relevant state variables and they are constant over time (Markov assumption); (2) players are forward looking, maximize expected intertemporal payoffs and know their own strategy functions; and (3) players’ strategies are common knowledge. The concept of rational behavior that I consider here maintains the assump-tions (1) and (2) of Markov strategy funcassump-tions and forward looking behavior. I relax con-dition (3). In our game, the payoff relevant state variables for player i are {Xt, εit} where Xt ≡ (Z1, Z2, Y1t−1,Y2t−1). Let σi(Xt, εit) be a strategy function for player i. Given a
strategy function σi(Xt, εit), we can define a conditional choice probability (CCP) function Pi(Xt) ≡
Z
I{σi(Xt, εit) = 1}dHi(εit), where I{.} is the indicator function. It will be
convenient to represent players’ behavior using CCPs. Player i’s beliefs about the expected behavior of his opponent can be represented as a CCP function Pj(Xt).
A strategy functionσi(Xt, εit) isrational if for every possible value of(Xt, εit) the action σi(Xt, εit)maximizes player i’s expected and discounted payoffs given playeri’s belief about
his opponent’s strategy. More formally, σi(Xt, εit)is arational strategy function if there is a
CCP function Pj(Xt)such that for any possible state (Xt, εit):
σi(Xt, εit) = ⎧ ⎨ ⎩ 1 if εit≤v Pj i (Xt) 0 otherwise (2)
where the functionvPj
i (Xt)represents the expected value offirmiif it is active today minus
its value if it is not active today given that: (1) current state is Xt; (2) firm i behaves
optimally in the future; and (3) firmi believes that its opponent’s CCP function is Pj. We
denote vPj
i as the differential value function. For given Pj, the function v Pj
i is implicitly
defined as the unique solution of a contraction mapping. I do not present here the details of the fixed point mapping that defines vPj
i . Assumption M and Lemmas 1 and 2, below,
present several properties of the function vPj
results. According to this definition of rational strategy, we say that a CCP function Pi is
rational if there is a CCP functionPj for the opponent such that for every valueX:
Pi(X) =Hi ³ vPj i (X) ´ (3)
AssumptionM, below, establishes a monotonicity property of the differential value func-tion in this game: if the opponent increases his probability of being active at some stateX, then the differential value vPj
i declines at any state. This monotonicity of the differential
value function with respect toPj implies that the empirical implications of rationalizability
can be represented in terms of bounds on players’ choice probabilities. αi ≤0is a necessary
condition for Assumption M to hold, but it is not sufficient.
ASSUMPTION M: Let PA
j andPjB be two CCP functions such that PjA(X) ≥ PjB(X) for
any value ofX. Then, vP
A j
i (X)≤v PB
j
i (X) for any value ofX.
Assumption M has several implications. Let use {Pj = 1} to denote the CCP function
with Pj(X) = 1 for every value of X. And let’s use {Pj = 0} to denote the CCP function
with Pj(X) = 0for every value of X. AssumptionM implies that for any CCPPj and any
value ofX: v{Pj=1} i (X) ≤ v Pj i (X) ≤ v {Pj=0} i (X) (4)
If player i beliefs that his opponent will be active in the market with probability one under any possible state, then this belief generates the lowest differential value and the lowest probability of being active for playeri. Similarly, if playeribelieves that he is a monopolist without threat of entry, then this belief generates the largest differential value and the highest probability of being active for player i. Therefore, if we do not know player i’s beliefs, we can say that Pi is consistent with rational behavior if for every value X:
Hi ³ v{Pj=1} i (X) ´ ≤Pi(X)≤Hi ³ v{Pj=0} i (X) ´ (5)
A CCP function Pi is rationalizable of level 2 if there is a probability functionPj, which
represents playeri’s belief about player j0s behavior, such that P
j’s rational behavior, and Pi maximizes firm i0s expected value given his belief Pj. More
formally, Pi is rationalizable of level 2 if there is a Pj such that for every X:
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ Hj ³ v{Pi=1} j (X) ´ ≤Pj(X)≤Hj ³ v{Pi=0} j (X) ´ and Pi(X) =Hi ³ vPj i (X) ´ (6)
We can use Assumption M to represent the restrictions of level-2 rationalizability as bounds on players’ CCPs. Let’s use PjL,1 and PjU,1 to denote the lower and upper bounds, respectively, on player j’s CCPs given level-1 rationality: i.e., PjL,1(X) ≡Hj
³ v{Pi=1} j (X) ´ andPjU,1(X)≡Hj ³ v{Pi=0} j (X) ´
. AssumptionM implies that for anyPj that satisfies player j’s level-1 rationalizability and for any value of X:
vP U,1 j i (X) ≤ v Pj i (X) ≤ v PjL,1 i (X) (7)
Therefore, if we do not know player i’s beliefs, we can say that a strategy Pi is consistent
with rationalizability of level-2 iff:
Hi µ vP U,1 j i (X) ¶ ≤Pi(X)≤Hi µ vP L,1 j i (X) ¶ (8)
It is straightforward to extend this result to level-k rationalizability. Under level-k ratio-nalizability a strategy Pi should be between a lower bound PiL,k and an upper bound P
U,k i
where these bounds can be obtained recursively as follows. For any k ≥1:
PiL,k(X) = Hi µ vP U,k−1 j i (X) ¶ PiU,k(X) = Hi µ vP L,k−1 j i (X) ¶ (9) with PjL,0 = 0 andP U,0 j = 1.
2.3
Identi
fi
cation under rationalizability
Suppose that we have a random sample of independent markets at some period t. For each market in the sample we observe a realization of the variables {Y1t, Y2t, Y1t−1, Y2t−1, Z1, Z2}.
Notice that the variablesZ1 and Z2 do not vary over time but they have sample variability
because they vary across markets. We are interested in using this sample to estimate the vector of structural parameters θ≡{ηi,γi, αi :i= 1,2}.
Let Xi be the vector of exogenous and predetermined explanatory variables (Zi, Yit−1).
And define βi ≡(ηi,γi)such that Xiβi ≡Ziηi+γiYit−1. Also, defineX ≡(X1, X2) and let
SX be the support of X. Consider the following conditions:
(C1) The variance-covariance matrixesV(X1) andV(X2) have full rank.
(C2) For any player i, there is a variableXi ⊂ Xi such that βi 6= 0 and
condi-tional on any value of the other variables inX, that we represent as(Xi(−), Xj),
the random variable{Xi |(Xi(−), Xj)} has unbounded support.
(C3) For any player i,αi ≤0and Assumption M holds.
These assumptions are standard exclusion restrictions and large-support conditions on X. The following Lemmas are used in the proofs of the identification results.
LEMMA 1: For any Pj, the function v Pj
i (X) is strictly increasing in Xiβi. Furthermore, limXiβi→+∞v
Pj
i (X) = +∞ andlimXiβi→−∞v
Pj
i (X) =−∞. LEMMA 2: For any Pj, the functionv
Pj
i (X) is strictly increasing inαi.
Let P0
i(X) be the true conditional probability function Pr(Yit = 1|Xt = X) in the
population. And let θ0 be the true value of θ in the population. Level-k rationalizability implies the following restrictions on choice probabilities: for i∈{1,2} and anyX ∈SX
PiL,k¡X, θ0¢ ≤P0 i (X)≤ P U,k i ¡ X, θ0¢ (10)
To prove point identification ofθ0 we should show that for any vectorθ 6=θ0 there are values of X ∈SX for which the previous inequality does not hold: i.e., either P
L,k i (X, θ)> P 0 i(X) orPiU,k(X, θ)< P0 i (X).
THEOREM 1 (Point identification under level-1 rationalizability). Suppose that conditions (C1)-(C3) hold and players are level-1 rational. Then, the parametersβ01 andβ02 are point-identified. The rest of the parameters may not be point-identified.
PROOF: Suppose thatθ is such thatβi 6=β0i . By AssumptionM,Lemma 1 and conditions (C1) and (C2), for any value of(Xi(−), Xj)we can alwaysfind a finite value of Xi , that we
denote byX∗ i , such thatv {Pj=1} i (Xi∗, Xi(−), Xj;θ) =v{ Pj=0} i (Xi∗, Xi(−), Xj;θ0). If βi > β 0 i ,
then for values of X with Xi > Xi∗ we have that by construction:
PiL,1(X, θ) =Hi ³ v{Pj=1} i (X, θ) ´ > Hi ³ v{Pj=0} i (X, θ 0 ) ´ ≥Pi0(X)
Similarly, if βi < β0i , then for values of X with Xi < Xi∗ we have that P L,1 i (X, θ) = Hi ³ v{Pj=1} i (X, θ) ´ > Hi ³ v{Pj=0} i (X, θ 0)´ ≥P0 i(X). Q.E.D.¥
THEOREM 2 (Point identification under level-2 rationalizability). Suppose that conditions (C1)-(C3) hold and that players are rational and they know that they are rational. Then, all the structural parameters in θ0 are point-identified.
PROOF: The proof goes through three cases which cover all the possible values of θ with
θ 6=θ0.
Case (i): Suppose thatθis such thatβi 6=β0i . By AssumptionM,Lemma 1 and conditions (C1) and (C2), for any value of (Xi(− ), Xj) we can always find a finite value of Xi , that
we denote byX∗ i , such that v PjU,1 i (Xi∗, Xi(−), Xj;θ) =v PjL,1 i (Xi∗, Xi(−), Xj;θ0). If βi > β0i ,
then for values of X with Xi > Xi∗ we have that by construction:
PiL,2(X, θ) =Hi µ vP U,1 j i (X, θ) ¶ > Hi µ vP L,1 j i (X, θ 0 ) ¶ =PiU,2(X, θ0)≥Pi0(X)
Similarly, if βi < β0i , then for values of X with Xi < Xi∗ we have that P L,2
i (X, θ) > PiU,2(X;θ0)≥P0
i (X).
Case (ii): Suppose that θ is such that (β1 =β01 ) and(β2 =β02 ) butβi(−) 6=β0i(−). By condition (C1), there should be some value of Xi(−) for which Xi(−)
¡
βi(−)−β0i(−)¢ > 0. Given this value ofXi(−), by condition (C2) we canfind a value of Xj (either large or small
enough, depending on the sign of β0j ) such that both P U,1 j (X, θ) (i.e., Hj ³ v{Pi=0} j (X, θ) ´ ) andPjL,1(X, θ 0 )(i.e., Hj ³ v{Pi=1} j (X, θ 0
have that: PiL,2(X, θ)'Hi ³ v{Pj=1} i (X, θ) ´ > Hi ³ v{Pj=1} i (X, θ 0 )´'PiU,2(X, θ0)≥Pi0(X)
where the inequality Hi
³ v{Pj=1} i (X, θ) ´ > Hi ³ v{Pj=1} i (X, θ 0 )´ holds becauseXi(−)βi(−) > Xi(−)β0i(−) and Lemma 1.
Case (iii): Suppose that θ is such that β1 = β01 and β2 = β02 but αi 6= α0i. Suppose that αi > α0i. By condition (C2) we can makeXjβ0j large enough such thatP
L,1
j (X, θ
0)
'1 and
PjU,1(X, θ)'1. Then, given thatαi > α0i, by Lemma 2 we have that: PiL,2(X, θ)'Hi ³ v{Pj=1} i (X, θ) ´ > Hi ³ v{Pj=1} i (X, θ 0)´ 'PiU,2(X, θ0)≥Pi0(X)
Suppose that αi < α0i. By condition (C2) we can make Xjβ0j large enough such that PjL,1(X, θ) ' 1 and PjU,1(X, θ0) ' 1. Then, given that αi < α0i, by Lemma 2 we have
that: PiU,2(X, θ)'Hi ³ v{Pj=1} i (X, θ) ´ < Hi ³ v{Pj=1} i (X, θ 0)´ 'PiL,2(X, θ0)≤Pi0(X) Q.E.D. ¥
References
[1] Aradillas-Lopez, A. and E. Tamer (2007): "The Identification Power of Equilibrium in Games," Journal of Business and Economic Statistics, forthcoming.
[2] Aumann, R. and A. Brandenburger (1995): "Epistemic Conditions for Nash Equilib-rium," Econometrica, 63, 1161-1180.
[3] Fudenberg, D. and J. Tirole (1991): "Game Theory". The MIT Press. Cambridge, Massa-chusetts.
[4] Haile, P. and E. Tamer (2003): "Inference with an Incomplete Model of English Auc-tions," Journal of Political Economy, 111, 1—51.
[5] Tamer, E. (2003): “Incomplete Simultaneous Discrete Response Model with Multiple Equilibria,” Review of Economic Studies, 70, 147-167.
